G.2 Contact Self-Energies

The surface GREEN's functions defined in (G.13) enable us to rewrite (G.11) in a form very similar to (4.11),

$\displaystyle {[E \ensuremath{{\underline{I}}} - \ensuremath{{\underline{H}}} -...
...uremath{{\underline{G}}}^\mathrm{r}_{DD} \ = \ \ensuremath{{\underline{I}}} \ ,$ (G.14)

where,

\begin{displaymath}\begin{array}{lllll} \ensuremath{{\underline{\Sigma}}}^\mathr...
... \ensuremath{{\underline{\Sigma}}}^\mathrm{r}_R \ . \end{array}\end{displaymath} (G.15)

All other elements of $ \ensuremath{{\underline{\Sigma}}}^\mathrm{r}_\mathrm{C}$ are zero. $ \ensuremath{{\underline{\Sigma}}}^\mathrm{r}_L$ and $ \ensuremath{{\underline{\Sigma}}}^\mathrm{r}_R$ are self-energies due to the left and right contacts, respectively, and $ \ensuremath{{\underline{t}}}_{DL}=\ensuremath{{\underline{t}}}_{LD}^\dagger$ and $ \ensuremath{{\underline{t}}}_{DR}=\ensuremath{{\underline{t}}}_{RD}^\dagger$. By following the same procedure one obtains the equation of motion for the lesser and greater GREEN's functions as [116]

$\displaystyle \ensuremath{{\underline{G}}}^\mathrm{\gtrless}_{DD} \ = \ \ensure...
...mathrm{r}_\mathrm{C} \right] \ \ensuremath{{\underline{G}}}^\mathrm{a}_{DD} \ ,$ (G.16)

where,

\begin{displaymath}\begin{array}{lllll} \ensuremath{{\underline{\Sigma}}}^\mathr...
...emath{{\underline{\Sigma}}}^\mathrm{\gtrless}_R \ . \end{array}\end{displaymath} (G.17)

Since the contacts are by definition in equilibrium, one obtains (Appendix D.1)

\begin{displaymath}\begin{array}{lll} \ensuremath{{\underline{g}}}^\mathrm{<}_{_...
... \ \ensuremath{{\underline{a}}}_{_{1,1}}\ f_{R} \ , \end{array}\end{displaymath} (G.18)

where $ \ensuremath{{\underline{a}}}=i(\ensuremath{{\underline{g}}}^\mathrm{r}-\ensure...
...line{g}}}^\mathrm{a})=-2\Im\mathrm{m}
[\ensuremath{{\underline{g}}}^\mathrm{r}]$ is the spectral function and $ f_{L(R)}$ is the FERMI factor of the left (right) contact. By defining the broadening function as

\begin{displaymath}\begin{array}{lllllll} \ensuremath{{\underline{\Gamma}}}_{C_{...
...D} & = & \ \ensuremath{{\underline{\Gamma}}}_{R}\ , \end{array}\end{displaymath} (G.19)

equation (G.17) can be rewritten as

\begin{displaymath}\begin{array}{lllll} \ensuremath{{\underline{\Sigma}}}^\mathr...
...+i\ \ensuremath{{\underline{\Gamma}}}_R \ f_{R} \ . \end{array}\end{displaymath} (G.20)

In a similar manner one can show that

\begin{displaymath}\begin{array}{lllll} \ensuremath{{\underline{\Sigma}}}^\mathr...
...\ensuremath{{\underline{\Gamma}}}_{R}\ (1-f_{R})\ . \end{array}\end{displaymath} (G.21)

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors